Unique factorization

Unique factorization is the property that every integer greater than one can be expressed as a product of primes in exactly one way, apart from the order of the factors.

Proof by induction: Proof by induction establishes statements about natural numbers from a base case and a step from each case to the next. Induction proves that every integer greater than one has a prime factorization.
Euclid: Euclid was an ancient Greek mathematician whose Elements organized geometry and number theory into deductive proofs. Books VII and IX contain classical results underpinning prime factorization.
Unique factorization domain: A unique factorization domain is an integral domain in which every nonzero nonunit factors uniquely into irreducibles, up to units and order. The integer theorem is the basic example of this broader algebraic property.
Fundamental theorem of arithmetic: Every integer greater than one is a product of primes, uniquely up to their order. The theorem licenses representing an integer by its prime exponents.
Existence proof: An existence proof establishes that at least one object satisfying a specified condition exists. Repeatedly splitting composite integers proves a prime factorization exists.
Elements: Elements is Euclid's mathematical treatise, a systematic presentation of geometry and number theory. Its number-theory books establish results about primes and divisibility.
Gaussian integers: The Gaussian integers are complex numbers of the form a + bi, where a and b are integers. They retain unique factorization despite allowing factors absent from the ordinary integers.
Integer factorization: Integer factorization expresses a composite integer as a product of smaller integers, usually primes. Algorithms seek the prime factors guaranteed to exist and be unique.
Uniqueness proof: A uniqueness proof establishes that an object satisfying specified conditions is the only such object. Euclid's lemma proves that two prime factorizations cannot differ.
Euclid's Elements, Book VII: Book VII of Euclid's Elements develops elementary number theory, including definitions and propositions about primes and divisibility. Its propositions include the divisibility tools used to prove uniqueness.

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